where \(k\)  is the Boltzmann constant, \(T\)  is equal to the temperature of the material in the case of non-degenerate semiconductors, or \(T\)  is equal to the Fermi temperature \(T\equiv T_F\)  in the case of degenerate semiconductors and metals [1]}. The first term in the right hand side accounts for the diffusion of the carriers (this term is justified in the framework of the local equilibrium approximation [2]}, [3]}). The second term describes the electric potential \(V\) . The third term \(\mu _{\updownarrow }^{ch}\)  is the ÔÕpure chemical potentialÕÕ that accounts for the spin-flip scattering, and is the main parameter for the description of giant magnetoresistance effects [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. In the following, we assume that \(\mu _{\updownarrow }^{ch}\)  is uniform in order to treat uniquely the spin-accumulation due to the SHE. The electric field reads \(\vec{\mathcal {E}} = - \vec{\nabla }V = \mathcal {E}_x^0 \, \vec{e}_x + \mathcal {E}_y \, \vec{e}_y \) , where the \(x\) -component \(\mathcal {E}_x^0\)  is constant (due to the invariance along \(x\) ), so that the Poisson law is reduced to \(\partial _y \mathcal {E}_y = q \delta n^Q/\epsilon \) . Note that the electric field does not depend on the spin-channel, so that the electric potential \(V\)  couples the two channels in Eq.(REF ).
We can re-write the Poisson law with the help of the chemical potential Eq.(REF ) as follow:
\(\nabla ^2 \mu _{\updownarrow } - \lambda _D^2 \, \frac{q n_0}{\epsilon }\, \nabla ^2 \ln {\left(\frac{n_{\updownarrow }}{n_0} \right)} + \frac{q \delta n^Q}{ \epsilon } = 0\) 
